2,575 research outputs found
Local Criteria for Triangulation of Manifolds
We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use
Complexity of 3-manifolds
We give a summary of known results on Matveev's complexity of compact
3-manifolds. The only relevant new result is the classification of all closed
orientable irreducible 3-manifolds of complexity 10.Comment: 26 pages, 7 figures, minor correction
Riemannian simplices and triangulations
We study a natural intrinsic definition of geometric simplices in Riemannian
manifolds of arbitrary dimension , and exploit these simplices to obtain
criteria for triangulating compact Riemannian manifolds. These geometric
simplices are defined using Karcher means. Given a finite set of vertices in a
convex set on the manifold, the point that minimises the weighted sum of
squared distances to the vertices is the Karcher mean relative to the weights.
Using barycentric coordinates as the weights, we obtain a smooth map from the
standard Euclidean simplex to the manifold. A Riemannian simplex is defined as
the image of this barycentric coordinate map. In this work we articulate
criteria that guarantee that the barycentric coordinate map is a smooth
embedding. If it is not, we say the Riemannian simplex is degenerate. Quality
measures for the "thickness" or "fatness" of Euclidean simplices can be adapted
to apply to these Riemannian simplices. For manifolds of dimension 2, the
simplex is non-degenerate if it has a positive quality measure, as in the
Euclidean case. However, when the dimension is greater than two, non-degeneracy
can be guaranteed only when the quality exceeds a positive bound that depends
on the size of the simplex and local bounds on the absolute values of the
sectional curvatures of the manifold. An analysis of the geometry of
non-degenerate Riemannian simplices leads to conditions which guarantee that a
simplicial complex is homeomorphic to the manifold
Incompressibility criteria for spun-normal surfaces
We give a simple sufficient condition for a spun-normal surface in an ideal
triangulation to be incompressible, namely that it is a vertex surface with
non-empty boundary which has a quadrilateral in each tetrahedron. While this
condition is far from being necessary, it is powerful enough to give two new
results: the existence of alternating knots with non-integer boundary slopes,
and a proof of the Slope Conjecture for a large class of 2-fusion knots. While
the condition and conclusion are purely topological, the proof uses the
Culler-Shalen theory of essential surfaces arising from ideal points of the
character variety, as reinterpreted by Thurston and Yoshida. The criterion
itself comes from the work of Kabaya, which we place into the language of
normal surface theory. This allows the criterion to be easily applied, and
gives the framework for proving that the surface is incompressible. We also
explore which spun-normal surfaces arise from ideal points of the deformation
variety. In particular, we give an example where no vertex or fundamental
surface arises in this way.Comment: 37 pages, 8 figures. V2: New remark in Section 9.1, additional
references; V3 Minor edits, to appear in Trans. Amer. Math. So
Generalized Sums over Histories for Quantum Gravity II. Simplicial Conifolds
This paper examines the issues involved with concretely implementing a sum
over conifolds in the formulation of Euclidean sums over histories for gravity.
The first step in precisely formulating any sum over topological spaces is that
one must have an algorithmically implementable method of generating a list of
all spaces in the set to be summed over. This requirement causes well known
problems in the formulation of sums over manifolds in four or more dimensions;
there is no algorithmic method of determining whether or not a topological
space is an n-manifold in five or more dimensions and the issue of whether or
not such an algorithm exists is open in four. However, as this paper shows,
conifolds are algorithmically decidable in four dimensions. Thus the set of
4-conifolds provides a starting point for a concrete implementation of
Euclidean sums over histories in four dimensions. Explicit algorithms for
summing over various sets of 4-conifolds are presented in the context of Regge
calculus. Postscript figures available via anonymous ftp at
black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps.Comment: 82pp., plain TeX, To appear in Nucl. Phys. B,FF-92-
Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes
In [7], a notion of constant scalar curvature metrics on piecewise flat
manifolds is defined. Such metrics are candidates for canonical metrics on
discrete manifolds. In this paper, we define a class of vertex transitive
metrics on certain triangulations of ; namely, the boundary
complexes of cyclic polytopes. We use combinatorial properties of cyclic
polytopes to show that, for any number of vertices, these metrics have constant
scalar curvature.Comment: 15 pages, 4 figure
Constructing Intrinsic Delaunay Triangulations of Submanifolds
We describe an algorithm to construct an intrinsic Delaunay triangulation of
a smooth closed submanifold of Euclidean space. Using results established in a
companion paper on the stability of Delaunay triangulations on -generic
point sets, we establish sampling criteria which ensure that the intrinsic
Delaunay complex coincides with the restricted Delaunay complex and also with
the recently introduced tangential Delaunay complex. The algorithm generates a
point set that meets the required criteria while the tangential complex is
being constructed. In this way the computation of geodesic distances is
avoided, the runtime is only linearly dependent on the ambient dimension, and
the Delaunay complexes are guaranteed to be triangulations of the manifold
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